Isoperimetric point

In geometry, the isoperimetric point is a triangle center — a special point associated with a plane triangle. The term was originally introduced by G.R. Veldkamp in a paper published in the American Mathematical Monthly in 1985 to denote a point $P$ in the plane of a triangle $△ ABC$ having the property that the triangles $△ PBC, △ PCA, △ PAB$ have isoperimeters, that is, having the property that[1][2]

{\begin{aligned}&{\overline {PB}}+{\overline {BC}}+{\overline {CP}},\\=\ &{\overline {PC}}+{\overline {CA}}+{\overline {AP}},\\=\ &{\overline {PA}}+{\overline {AB}}+{\overline {BP}}.\end{aligned}}

Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions. The isoperimetric point of $△ ABC$ in the sense of Veldkamp, if it exists, has the following trilinear coordinates.[3]

{\displaystyle \sec {\tfrac {A}{2}}\cos {\tfrac {B}{2}}\cos {\tfrac {C}{2}}-1\

Given any triangle $△ ABC$ one can associate with it a point $P$ having trilinear coordinates as given above. This point is a triangle center and in Clark Kimberling's Encyclopedia of Triangle Centers (ETC) it is called the isoperimetric point of the triangle $△ ABC$ . It is designated as the triangle center X(175).[4] The point X(175) need not be an isoperimetric point of triangle $△ ABC$ in the sense of Veldkamp. However, if isoperimetric point of triangle $△ ABC$ in the sense of Veldkamp exists, then it would be identical to the point X(175).

The point $P$ with the property that the triangles $△ PBC, △ PCA, △ PAB$ have equal perimeters has been studied as early as 1890 in an article by Emile Lemoine.[4][5]

Existence of isoperimetric point in the sense of Veldkamp

A triangle

△ ABC

in which the triangle center X(175) is not the isoperimetric point in the sense of Veldkamp.

Let $△ ABC$ be any triangle. Let the sidelengths of this triangle be $a, b, c$ . Let its circumradius be $R$ and inradius be $r$ . The necessary and sufficient condition for the existence of an isoperimetric point in the sense of Veldkamp can be stated as follows.[1]

The triangle

△ ABC

has an isoperimetric point in the sense of Veldkamp if and only if

a+b+c>4R+r.

For all acute angled triangles $△ ABC$ we have $a + b + c > 4 R + r$ , and so all acute angled triangles have isoperimetric points in the sense of Veldkamp.

Properties

Let $P$ denote the triangle center X(175) of triangle $△ ABC$ .[4]

$P$ lies on the line joining the incenter and the Gergonne point of $△ ABC$ .
If $P$ is an isoperimetric point of $△ ABC$ in the sense of Veldkamp, then the excircles of triangles $△ PBC, △ PCA, △ PAB$ are pairwise tangent to one another and $P$ is their radical center.
If $P$ is an isoperimetric point of $△ ABC$ in the sense of Veldkamp, then the perimeters of $△ PBC, △ PCA, △ PAB$ are equal to

{\frac {2\triangle }{{\bigl |}4R+r-(a+b+c){\bigr |}}}

where $△$ is the area, $R$ is the circumradius, $r$ is the inradius, and $a, b, c$ are the sidelengths of $△ ABC$ .[6]

Soddy circles

Inner and outer Soddy circles in the case where the outer Soddy point is an isoperimetric point in the sense of Veldkamp.

Inner and outer Soddy circles in the case where the outer Soddy point is not an isoperimetric point in the sense of Veldkamp.

Given a triangle $△ ABC$ one can draw circles in the plane of $△ ABC$ with centers at $A, B, C$ such that they are tangent to each other externally. In general, one can draw two new circles such that each of them is tangential to the three circles with $A, B, C$ as centers. (One of the circles may degenerate into a straight line.) These circles are the Soddy circles of $△ ABC$ . The circle with the smaller radius is the inner Soddy circle and its center is called the inner Soddy point or inner Soddy center of $△ ABC$ . The circle with the larger radius is the outer Soddy circle and its center is called the outer Soddy point or outer Soddy center of triangle $△ ABC$ . [6][7]

The triangle center X(175), the isoperimetric point in the sense of Kimberling, is the outer Soddy point of $△ ABC$ .

References

G. R. Veldkamp (1985). "The isoperimetric point and the point(s) of equal detour". Amer. Math. Monthly. 92 (8): 546–558. doi:10.2307/2323159. JSTOR 2323159.
Hajja, Mowaffaq; Yff, Peter (2007). "The isoperimetric point and the point(s) of equal detour in a triangle". Journal of Geometry. 87 (1–2): 76–82. doi:10.1007/s00022-007-1906-y. S2CID 122898960.
Kimberling, Clark. "Isoperimetric Point and Equal Detour Point". Retrieved 27 May 2012.
Kimberling, Clark. "X(175) Isoperimetric Point". Archived from the original on 19 April 2012. Retrieved 27 May 2012.
The article by Emile Lemoine can be accessed in Gallica. The paper begins at page 111 and the point is discussed in page 126.Gallica
Nikolaos Dergiades (2007). "The Soddy Circles" (PDF). Forum Geometricorum. 7: 191–197. Retrieved 29 May 2012.
"Soddy Circles". Retrieved 29 May 2012.

External links

isoperimetric and equal detour points - interactive illustration on Geogebratube

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[Veldkamp-1] G. R. Veldkamp (1985). "The isoperimetric point and the point(s) of equal detour". Amer. Math. Monthly. 92 (8): 546–558. doi:10.2307/2323159. JSTOR 2323159.

[2] Hajja, Mowaffaq; Yff, Peter (2007). "The isoperimetric point and the point(s) of equal detour in a triangle". Journal of Geometry. 87 (1–2): 76–82. doi:10.1007/s00022-007-1906-y. S2CID 122898960.

[3] Kimberling, Clark. "Isoperimetric Point and Equal Detour Point". Retrieved 27 May 2012.

[ETC-4] Kimberling, Clark. "X(175) Isoperimetric Point". Archived from the original on 19 April 2012. Retrieved 27 May 2012.

[5] The article by Emile Lemoine can be accessed in Gallica. The paper begins at page 111 and the point is discussed in page 126.Gallica

[Dergi-6] Nikolaos Dergiades (2007). "The Soddy Circles" (PDF). Forum Geometricorum. 7: 191–197. Retrieved 29 May 2012.

[7] "Soddy Circles". Retrieved 29 May 2012.